What Would a Flatlander Really See?

A direct link to this video is at http://www.youtube.com/watch?v=73IGTygl_Q4

One of the questions that sometimes comes up about the original Imagining the Tenth Dimension animation is that it doesn't really show what a Flatlander's world would look like to a Flatlander. The concept of 2D creatures living in a flat, two-dimensional world was first introduced to us by Edwin A. Abbott in his 1884 novel "Flatland: a Romance of Many Dimensions".

In the original animation, we started out by imagining our flatlander as being the "one-eyed Jack on an impossibly flat playing card":



This is the original image Jason Orban of OH!Media drew for me as a representation of what I'm describing here. But why did we say "impossibly flat" in the narration? Because no matter what kind of paper we can imagine this Flatlander to be drawn upon, that paper has length and width and some small amount of depth, represented by the thickness of that paper. For us to really imagine a flatlander we need to be thinking about him in a plane that has length and width only, and no depth whatsoever. This is not an easy thing for we 3D observers to imagine, and some people will insist that there must be at least some depth for this flatlander to exist. It's important to realize what we're talking about here: in the same way that a one dimensional line must have length only, no width or depth, for it to truly be one-dimensional, a 2D flatlander must have length and width only, and no depth, or he is not really two-dimensional.

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Each additional dimension adds an additional degree of freedom. If you lived as a point on a one-dimensional line, the only thing you would be able to see would be the two points that are nearest you on your line. Everything else would be hidden from view, and because you existed within a single dimension, there would be no way for you to see or travel "around" those points to become aware of what lies beyond. In fact, the concept of going "around" would be completely unimaginable to you, and you would just take it for granted that your point of view was the only point of view possible.

Adding a second dimension gives our flatlander a little more room to move and see "around" other objects, but the possibilities for this creature, living within a flat 2D plane, are still very limited when you compare them to our world: as we watch the flatlander from above, we are able to see things with much more freedom that what his perspective would really allow. Let's look a little closer at that one-eyed-Jack as he was shown in the animation:



Here's a question we can ask ourselves then: are we seeing the flatlander's left eye or right eye in this picture? From our 3D perspective, we look down upon the flatlander as pictured here and presume that his eye is to the right of his nose. But really, because he is completely flat, this flatlander's eye is "behind" his nose, which means the only thing this poor fellow would be able to see is the inside of his own head! We would need to move his eye to the outside edge of his face if we really wanted to let him see the world around him, like so:

Again, from our imaginary flatlander's perspective, having to have your eyes on the "outside edge" would seem completely normal, and the idea of an eye "within" a head would be completely foreign. But what would his 2D eye really be able to see? We still haven't properly imagined his perspective. As we try to do that, let's think about what we really see as 3D creatures.

Most of us have two eyes. Let's start by covering one eye, so that we're more like the flatlander we're looking at here. If you sit motionless with one eye covered and look in front of you, how many dimensions are you seeing? Most of us would automatically say three (or even four if we consider the direction of time), but really we are only seeing two: we are seeing a flat 2D representation of the world in front of us, projected onto the retina at the back of our eye. If we keep our other eye covered and move our heads around a bit, we'll see changing relationships within that 2D image that give us clues as to what is nearer and what is further away from us, but that is information that our brains are deducing from watching a series of 2D images our eye is delivering to the brain, one after the other. Johnny Lee has a lovely YouTube video showing a "heads up" display he created using Nintendo Wii technology that allows a flat image on a screen to appear as if it has a very realistic 3D depth, and his invention relies upon exactly what we're talking about here: our brains are able to look at a flat 2D image and infer the 3D nature of that image by what happens as we move our heads around.

A direct link to this video is at http://www.youtube.com/watch?v=Jd3-eiid-Uw

The great thing about the Johnny Lee demo is it clearly shows how our brains can deduce 3D even with only one eye, as long as we can move our heads around. If you add a second eye to our 2d flatlander, or to us as 3D creatures, you give the brain even more information to work with - now there are differences between what one eye and the other sees, and the differences get greater the closer any objects are to us, which allows the brain to infer even more about the 3D nature of the world it's witnessing... but our brains do so, still, by taking in two slightly different 2D images projected on the back of our eyeballs. This convinces us that we are able to see in 3D, but the truth is still that our perception of 3D is happening within our brains, and the information that is coming into each eye is still really only a series of changing 2D images.

So: our 3D eyes are seeing 2D images, and we take it for granted that we can't see what's on the other side of a wall without getting up and moving there. An imaginary one-dimensional creature would only be able to see the Zero-D "points" that are nearest them on their 1D line, and to them the idea of seeing or moving "around" things would be inconceivable, because of the extremely limited freedom of movement their dimension gives them.

In each case, what each creature sees from its current dimension is the dimension below. Now we're ready to go back to our original question: what would a flatlander really see?

Try to imagine yourself now as a 2D flatlander. In our representation of him, we're seeing his world from above - we're able to see "around" things that the flatlander would not. In fact, until we repositioned his eyeball he would not even have been able to see "around" the outline of his own face. What we need to do now is imagine ourselves rotating down out of our nice round 3D world, and becoming part of the flat 2D plane the flatlander exists within. To do that, we need to rotate down into his perspective. This series of pictures shows us going most of the way:
This sequence shows what happens as you get closer and closer to the perspective of the flat 2D plane our flatlander is living within. You would get a similar effect by picking up a piece of paper, bringing it up to eye level and rotating it so that you're only seeing one edge. The closer you get to seeing just that edge, the closer you are to seeing what the flatlander would see - but it's still important to keep in mind that the edge of a piece of a piece of paper in our 3D world is still a whole dimension thicker than the forest of lines, some nearer, some further away, some hidden behind other lines that would be the perceived 2D world of the flatlander.

The original eleven-minute animation for Imagining the Tenth Dimension introduces viewers to the ideas from chapter one of the book of the same name. In the book I say this:

To a Flatlander, we 3D beings would be able to pop in and out of their two-dimensional world as if by magic, and our texture and form would be quite inexplicable. In the same way, we humans would find the 2D information that a Flatlander sees to be a useless and confusing jumble of lines all in the same plane.

If you go back and watch the original animation now, keep in mind that what we're seeing in the section about flatlanders is all from our perspective, looking down on the flatlander's world from "above", which is a direction the flatlander wouldn't even be able to conceive of. Most books introducing people to the idea of flatlanders choose to use this vantage point, and for good reason: because what a flatlander would really see is even stranger than what we showed in our brief introduction to thinking about the dimensions.

Finally, please kind in mind that what we're talking about here with Imagining the Tenth Dimension is a visualization tool, not a scientific proof. But as a visualization, it does gives us ways to imagine lots of things that physicists say about our reality, for instance:

• how our experience of the fifth dimension is curled up at the planck length (because time is a direction in the fourth spatial dimension and the fifth dimension's probabilistic outcomes are being observed one planck length after another)
• the string theory idea that our 3D universe is "locked in" at the seventh dimension by a 3D and 7D brane
• the idea that by the time you get to the tenth dimension you are visualizing the equilibrium state that exists "outside the system" as Godel liked to describe it. Physicist Sean Carroll talks about this "equilibrium state" that exists before and after our observed universe in the June 2008 issue of Scientific American. Other people refer to this state as the "omniverse".
  
• Everett's Many Worlds Interpretation of quantum mechanics, which says there are branching parallel universes being created through choice and chance (this idea was confirmed by a proof offered in September 2007 by a team of scientists at Oxford under the direction of physicist David Deutsch), and
• the idea that the future exists only as part of probabilistic waves, as proven by experiments conducted by a team of scientists in Vienna under the direction of physicist Anton Zeilinger.

What's unique about this new way of thinking about time and space is that it makes no distinction between the first four dimensions that we know from spacetime, and the extra dimensions that physicists are talking about: this project proposes that since all of the ten dimensions are spatial dimensions, each new dimension adds an additional degree of freedom, and we can use what we know about the dimensions below to help us visualize the dimensions beyond the four that we live in.

There are lots of other discussions from physics and philosophy that can stem from this way of visualizing how our reality is constructed and what that means to us, I do hope you enjoy the journey as we explore those ideas.

From Imagining the Tenth Dimension,

Rob Bryanton

Related blog entries:
Flatlanders on a Line
Time is a Direction
The Omniverse
Hypercubes and Plato's Cave
Time in Either Direction
Anime, Gaming and Cusps

Next: What Would a Linelander Really See?